Point Collocation Method Research Articles

Damage analysis and dynamic response of elasto-plastic laminated composite shallow spherical shell under low velocity impact

Based on the elasto-plastic mechanics, the damage analysis and dynamic response of an elasto-plastic laminated composite shallow spherical shell under low velocity impact are carried out in this paper. Firstly, a yielding criterion related to spherical tensor of stress is proposed to model the mixed hardening orthotropic material, and accordingly an incremental elasto-plastic damage constitutive relation for the laminated shallow spherical shell is founded when a strain-based Hashin failure criterion is applied to assess the damage initiation and propagation. Secondly, using the presented constitutive relations and the classical nonlinear shell theory, a series of incremental nonlinear motion equations of orthotropic moderately thick laminated shallow spherical shell are obtained. The questions are solved by using the orthogonal collocation point method, Newmark method and iterative method synthetically. Finally, a modified elasto-plastic contact law is developed to determine the normal contact force and the effect of damage, geometrical parameters, elasto-plastic contact and boundary conditions on the contact force and the dynamic response of the structure under low velocity impact are investigated.

An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

The aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to Dirichlet, Neumann and Robin boundary conditions. Toward this, the meshless point collocation method (MPCM) is used for MHD flow problems in channels with fully insulating or partially insulating and partially conducting walls, having rectangular, circular, elliptical or even arbitrary cross sections. MPC is a truly meshless and computationally efficient method. The maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently, and the convergence proof for the numerical solution of the Poisson problem with Dirichlet boundary conditions have been attained also. Additionally, in the present work convergence is attained for Neumann and Robin boundary conditions, accordingly. The shape functions are constructed using the Moving Least Squares (MLS) approximation. The refinement procedure with meshless methods is obtained with an easily handled and fully automated manner. We present results for Hartmann number up to 10 5 . The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional FEM and BEM, for the Dirichlet, Neumann and Robin boundary conditions of interior problems with simple or complex boundaries.

Adaptive support domain implementation on the Moving Least Squares approximation for Mfree methods applied on elliptic and parabolic PDE problems using strong-form description

The extent of application of meshfree methods based on point collocation (PC) techniques with adaptive support domain for strong form Partial Differential Equations (PDE) is investigated. The basis functions are constructed using the Moving Least Square (MLS) approximation. The weak-form description of PDEs is used in most MLS methods to circumvent problems related to the increased level of resolution necessary near natural (Neumann) boundary conditions (BCs), dislocations, or regions of steep gradients. Alternatively, one can adopt Radial Basis Function (RBF) approximation on the strong-form of PDEs using meshless PC methods, due to the delta function behavior (exact solution on nodes). The present approach is one of the few successful attempts of using MLS approximation [Atluri, Liu, and Han (2006), Han, Liu, Rajendran and Atluri (2006), Atluri and Liu (2006)] instead of RBF approximation for the meshless PC method using strong-form description. To increase the accuracy of the MLS interpolation method and its robustness in problems with natural BCs, a suitable support domain should be chosen in order to ensure an optimized area of coverage for interpolation. To this end, the basis functions are constructed using two different approaches, pertinent to the dimension of the support domain. On one hand, a compact form for the support domain is retained by keeping its radius constant. On the other hand, one can control the number of neighboring nodes as the support domain of each point. The results show that some inaccuracies are present near the boundaries using the first approach, due to the limited number of nodes belonging to the support domain, which results in failed matrix inversion. Instead, the second approach offers capa1 Department of Medical Physics, School of Medicine, University of Patras, GR 26500, Rion, Greece. 2 Department of Chemical Engineering, University of Patras, GR 26500, Rion, Greece. 3 Institute of Chemical Engineering and High Temperature Chemical Processes Foundation for Research and Technology, P.O. Box 1414, GR-26504, Patras, Greece. 4 Correspondence to Eugene.Skouras@iceht.forth.gr 2 Copyright © 2009 Tech Science Press CMES, vol.43, no.1, pp.1-25, 2009 bility for fully matrix inversion under many (if not all) circumstances, resulting in basis functions of increased accuracy and robustness. This PC method, applied along with an intelligent adaptive refinement, is demonstrated for elliptic and for parabolic PDEs, related to many flow and mass transfer problems.

On the Truly Meshless Solution of Heat Conduction Problems in Heterogeneous Media

A truly meshless method based on the weighted least-squares (WLS) approximation and the method of point collocation is proposed to solve heat conduction problems in heterogeneous media. It is shown that, in the case of strong heterogeneity, accurate and smooth solutions for temperature and heat flux can be obtained by applying the WLS approximation in each homogeneous domain and using a double-stage WLS approximation technique together with a proper neighbor selection criterion at each interface.

A Comparison of Some Model Order Reduction Methods for Fast Simulation of Soft Tissue Response using the Point Collocation-based Method of Finite Spheres (PCMFS).

In this paper we develop the Point Collocation-based Method of Finite Spheres (PCMFS) to simulate the viscoelastic response of soft biological tissues and evaluate the effectiveness of model order reduction methods such as modal truncation, Hankel optimal model and truncated balanced realization techniques for PCMFS. The PCMFS was developed in [1] as a physics-based technique for real time simulation of surgical procedures. It is a meshfree numerical method in which discretization is performed using a set of nodal points with approximation functions compactly supported on spherical subdomains centered at the nodes. The point collocation method is used as the weighted residual technique where the governing differential equations are directly applied at the nodal points. Since computational speed has a significant role in simulation of surgical procedures, model order reduction methods have been compared for relative gains in efficiency and computational accuracy. Of these methods, truncated balanced realization results in the highest accuracy while modal truncation results in the highest efficiency.

Free vibration of antisymmetric angle-ply-laminated plates including transverse shear deformation: Spline method

A free vibration study of antisymmetric angle-ply composite plates including shear deformation and rotatory inertia using the point collocation method and applying spline function approximations is presented. The equations of motion for the plate are derived using the theory of Yang, Norris and Stavsky. The solution is assumed in a separable form to obtain a system of coupled differential equations in displacement and rotational functions and these functions were approximated by Bickley-type splines of order three. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibrations of two- and four-layered plates, made up of several types of layer materials and subjected to two types of boundary conditions are considered. Parametric studies were made of the variation of frequency parameters with respect to the aspect ratio, side-to-thickness ratio and ply angle. The numerical results are presented through diagrams and, in some cases, are compared with results obtained by FEM.

An efficient method to compute the dynamic response of a fluid-filled crack

SUMMARY The vibration of a fluid-filled crack is considered to be one of the most plausible source mechanisms for the long-period events and volcanic tremors occurring around volcanoes. As a tool for the quantitative interpretation of source process of such volcanic seismic signals, we propose a method to numerically simulate the dynamic response of a fluid-filled crack. In this method, we formulate the motions of the fluid inside and the elastic solid outside of the crack, using boundary integrals in the frequency domain and solve the dynamic interactions between the fluid and the elastic solid using the point collocation method. The present method is more efficient compared with the time-domain finite difference method, which has been used in simulations of a fluid-filled crack and enables us to study the dynamics of a fluid-filled crack over a wide range of physical parameters. The method also allows us direct calculation of the attenuation quality factor of the crack resonance, which is an indispensable parameter for estimating the properties of the fluid inside the crack. The method is also designed to be flexible to many applications, which may be encountered in volcano seismology, and thus, extensions of the method to more complicated problems are promising.

A Forging Simulation by Using the Point Collocation Method with a Boundary Layer of Finite Element

Point collocation methods have no mesh, no integration. While, the robustness of the point collocation methods is an issue especially when scattered and random points are used. To improve the robustness, some studies suggest that the positivity conditions can be important when using the point collocation methods. For boundary points, however, the positivity conditions cannot be satisfied, so that it is possible to get large numerical errors from the boundary points when using the point collocation methods. The author has proposed a point collocation method with a boundary layer of finite element. In this method, by introducing a boundary layer of finite element in boundary domain of workpiece, unsatisfactory issue of the positivity conditions of boundary points can be avoided, and the complicated boundary conditions can be easily imposed with the boundary layer of finite element. A forging process is analyzed by using the point collocation method with a boundary layer of finite element.

A least-square radial point collocation method for adaptive analysis in linear elasticity

This paper presents a least-square radial point collocation method (LS-RPCM) that is formulated based on the strong formulation and the local approximation using radial basis functions (RBFs). Aiming to solve the instability problem observed in the conventional RPCM using local nodes, a simple and yet effective procedure that uses the well-known least-square technique in a carefully designed manner has been proposed to restore the stability. Since stable solution can now be obtained, the LS-RPCM is then extended for adaptive analysis. Attractive features of the meshfree strong-form method that facilitate the implementation of adaptive analysis are demonstrated via a number of examples in this work. A robust residual based error estimator and a simple refinement procedure using Delaunay diagram are adopted in our adaptive scheme. Stable and accurate results are obtained in all the numerical examples.

Meshfree point collocation method for the stream-vorticity formulation of 2D incompressible Navier–Stokes equations

Meshfree point collocation method is developed for the stream-vorticity formulation of two-dimensional incompressible Navier–Stokes equations. Particular emphasis is placed on the novel formulation of effective vorticity condition on no-slip boundaries. The moving least square approximation is employed to construct shape functions in conjunction with the framework of point collocation method. The derivatives of an arbitrary function can be obtained by the linear combination of these shape functions, which enables the vorticity boundary condition to vary linearly with the boundary velocity and the stream function. Together with the second-order partial differential equations for the stream and vorticity functions, this boundary condition of vorticity provides the efficient meshfree point collocation scheme for the two-dimensional incompressible flow. The accuracy and stability for the proposed scheme are demonstrated through a new type of application problem with complex geometry, in addition to several typical examples of steady-state flow simulation.

Meshfree point collocation method with intrinsic enrichment for interface problems

A meshfree collocation method with an intrinsic wedge enrichment is presented for solving interface problems. To approximate the class of functions with discontinuous derivatives on the interface, the wedge is asymptotically added to the basis functions. A general class of wedge basis functions with specified orders of asymptotic behavior at the interface is developed for moving least square approximations. These are implemented in diffuse derivative methods where the shape functions are approximately differentiated. The reproducing properties of these approximations for the polynomial part and for the wedge function along straight boundaries of the basis are demonstrated. For curved boundaries, the reproducing properties of the wedge functions are more restricted. Numerical results show the ease of constructing the intrinsic enrichment and the robustness of the numerical scheme in solving interface problems.

A subdomain collocation method based on Voronoi domain partition and reproducing kernel approximation

A subdomain collocation reproducing kernel approximation method is proposed. Subdomains are constructed by nonoverlapping Voronoi cells and a local weak form is defined over these subdomains. The standard RKPM shape functions are used directly for approximation, while weighting functions of the subdomain collocation method hold a constant value of unit only over a specific Voronoi cell. The body integration of the local weak form is now converted into much cheaper and efficient contour integration along the boundaries of Voronoi cells. It does not need to impose traction free boundary conditions explicitly in contrast to point collocation method. Furthermore, the method provides a natural background structure for performing h-adaptivity analysis straightforwardly. All these features constitute the subdomain collocation method a promising alternative to standard Galerkin method and point collocation method. Some elastostatics examples are presented to demonstrate the effectiveness, the convergence property, and the adaptivity performance of present method.

A regularized least-squares radial point collocation method (RLS-RPCM) for adaptive analysis

This paper presents a stabilized meshfree method formulated based on the strong formulation and local approximation using radial basis functions (RBFs). The purpose of this paper is two folds. First, a regularization procedure is developed for stabilizing the solution of the radial point collocation method (RPCM). Second, an adaptive scheme using the stabilized RPCM and residual based error indicator is established. It has been shown in this paper that the features of the meshfree strong-form method can facilitated an easier implementation of adaptive analysis. A new error indicator based on the residual is devised and used in this work. As shown in the numerical examples, the new error indicator can reflect the quality of the local approximation and the global accuracy of the solution. A number of examples have been presented to demonstrate the effectiveness of the present method for adaptive analysis.

Axisymmetric Nonlinear Stability of a Shallow Conical Shell with a Spherical Cap of Arbitrary Variable Shell Thickness

The axisymmetric nonlinear stability of a shallow conical shell with a spherical cap was studied using the point collocation method with the cubic B-spline function as a trial function. Formulas were set up to consider arbitrary variable shell thickness and different boundary and loading conditions. A FORTRAN program was written to determine the critical loads and trace the stable parts of the equilibrium paths of the shell by method of gradually applied load. The upper and lower critical loads obtained in this technical note for some specific cases with constant shell thickness are of very good accuracy and agree very well with known solutions and finite-element method results. The method was then expanded successfully to arbitrary variable shell thickness and different boundary and loading conditions.

Simulation of an extruded quadrupolar dielectrophoretic trap using meshfree approach

A numerical simulation of an extruded quadrupolar dielectrophoretic (DEP) trap is presented in this paper. To compute the electric field distribution of the trap, a meshfree collocation technique, termed as radial point collocation method (RPCM), is formulated. The simulation of the particle behaviors in the DEP trap is conducted by superimposing the DEP simulation results obtained using RPCM and the hydrodynamic results obtained using a commercial CFD package. The present model has been validated by experimental data and simulation results obtained using other numerical methods. The model is able to determine the full force field within the trap domain and to predict the trap strength under different electrode voltages. The present model has been used for parametric study of the extruded quadruplar DEP trap. Meshfree technique used in this work has apparent advantages over traditional finite element method during the design optimization process because of the absence of remeshing.

Effects of numerical formulation on magnetic field computation using meshless methods

Meshless methods have some advantages over their counterparts such as the finite-element method (FEM). However, existing meshless methods for computational electromagnetic fields are still not as efficient as FEM. In this paper, we compare two meshless methods of discretizing the computational domain of Poisson-like problems; namely, the point collocation and Galerkin methods (which use the strong and weak forms of the governing equation respectively), and their effects on the computational accuracy and efficiency of the magnetic fields. We also discuss methods of handling discontinuities at the material interface. We present several examples, which also provide a means to validate and evaluate both meshless methods. Exact solutions and/or FEM are used as a basis for comparison. In addition, we also verify the results by comparing computed magnetic forces against those measured experimentally

Extrinsic meshfree approximation using asymptotic expansion for interfacial discontinuity of derivative

A sharp meshfree approximation for derivative discontinuities across arbitrary interfaces is proposed. The interface can be arbitrarily located in a domain in which nodes are distributed uniformly or irregularly. The proposed meshfree approximations consist of two parts, singular and regular. The moving least square meshfree approximation is used together with the local wedge function as basis functions. The approximations for discontinuities are applied in a meshfree point collocation method to obtain solutions of the Poisson problem with a layer delta source on the interface and second order elliptic problems with discontinuous coefficients and/or the singular layer sources along the interface. The numerical calculations show that this method has good performance even on irregular node models.

Radial point collocation method (RPCM) for solving convection-diffusion problems

In this paper, Radial point collocation method (RPCM), a kind of meshfree method, is applied to solve convection-diffusion problem. The main feature of this approach is to use the interpolation schemes in local supported domains based on radial basis functions. As a result, this method is local and hence the system matrix is banded which is very attractive for practical engineering problems. In the numerical examination, RPCM is applied to solve non-linear convection-diffusion 2D Burgers equations. The results obtained by RPCM demonstrate the accuracy and efficiency of the proposed method for solving transient fluid dynamic problems. A fictitious point scheme is adopted to improve the solution accuracy while Neumann boundary conditions exist. The meshfree feature of the present method is very attractive in solving computational fluid problems.

Three dimension profile extrusion simulation using meshfree method

A meshfree method based on reproducing kernel approximation and point collocation is presented for analysis of 3D profile extrusion processes. The point collocation method is a true meshfree method without the employment of a background mesh. We show that, in a point collocation approach, the implementation of program does not need the background mesh, which is a very time consuming process in a Galerkin method for integration. Among the points of the meshfree model, there is no connecting information. A mesh quality control method with mold interpenetration checking based on the Delaunay Bowyer-Watson algorithm is introduced to produce the topological relations, which enable us to show meshfree simulation results using the same procedure as in the finite element method. A C-shape profile extrusion application example is presented and compared with the finite element method to verify the proposed method.

Point interpolation collocation method for the solution of partial differential equations

This paper presents a truly meshless method for solving partial differential equations based on point interpolation collocation method (PICM). This method is different from the previous Galerkin-based point interpolation method (PIM) investigated in the papers [G.R. Liu, (2002), mesh free methods, Moving beyond the Finite Element Method, CRC Press. G.R. Liu, Y.T. Gu, A point interpolation method for two-dimension solids, Int J Numer Methods Eng, 50, 937–951, 2001. G.R. Liu, Y.T. Gu, A matrix triangularization algorithm for point interpolation method, in Proceedings Asia-Pacific Vibration Conference, Bangchun Weng Ed., November, Hangzhou, People's Republic of China, 2001a, 1151–1154. 1–3.], because it is based on collocation scheme. In the paper, polynomial basis functions have been used. In addition, Hermite-type interpolations called as inconsistent PIM has been adopted to solve PDEs with Neumann boundary conditions so that the accuracy of the solution can be improved. Several examples were numerically analysed. These examples were applied to solve 1D and 2D partial differential equations including linear and non-linear in order to test the accuracy and efficiency of the presented method based on polynomial basis functions. The h-convergence rates were computed for the PICM based on different model of regular and irregular nodes. The results obtained by polynomial PICM show the presented schemes possess a considerable perfect stability and good numerical accuracy even for scattered models while matrix triangularization algorithm (MTA) adopted in the computed procedure.